Conformal anomaly, entanglement entropy and boundaries
Sergey N. Solodukhin University of Tours
Talk at Oxford Holography Seminar February 9, 2016 Plan of the talk:
1. Brief review: local Weyl anomaly, entangle- ment entropy
2. Integral Weyl anomaly in presence of bound- aries
a) d=4 b) d=6
3. Integral Weyl anomaly in odd dimensions
4. Entanglement entropy and boundaries
5. Some open questions
1 Based on
1. S.S., “Boundary terms of conformal anomaly,” Phys. Lett. B 752, 131 (2016) [arXiv:1510.04566 [hep-th]].
2. Dima Fursaev and S.S. “Anomalies, entropy and boundaries,” arXiv:1601.06418 [hep-th].
3. work in progress with Amin Astaneh, Clement Berthiere
2 Other recent relevant works:
1. D. Fursaev, “Conformal anomalies of CFT’s with boundaries,” arXiv:1510.01427 [hep-th]
2. C. P. Herzog, K. W. Huang and K. Jensen, “Universal Entanglement and Boundary Geom- etry in Conformal Field Theory,” arXiv:1510.00021 [hep-th].
3 Earlier relevant works:
1. J. S. Dowker and J. P. Schofield, “Confor- mal Transformations and the Effective Action in the Presence of Boundaries,” J. Math. Phys. 31, 808 (1990).
2. J. Melmed, “Conformal Invariance and the Regularized One Loop Effective Action,” J. Phys. A 21, L1131 (1988).
3. I. G. Moss, “Boundary Terms in the Heat Kernel Expansion,” Class. Quant. Grav. 6, 759 (1989).
4. T. P. Branson, P. B. Gilkey and D. V. Vas- silevich, “The Asymptotics of the Laplacian on a manifold with boundary. 2,” Boll. Union. Mat. Ital. 11B, 39 (1997)
4 Let me first remind you briefly the standard story
5 Local Weyl anomaly
ν c g T ν = R, d = 2 24π
ν a b 2 g T ν = E4+ Tr W , d = 4 −5760π2 1920π2
2 αβ ν ν 1 2 Tr W = R R 2R νR + R αβ ν − 3
αβ ν ν 2 E = R R 4R νR + R . 4 αβ ν − (For scalar field a = b = 1)
ν g T ν = 0 , d = 2n + 1
6 Entanglement entropy and Weyl anomaly
Σ is compact 2d entangling surface
A(Σ) S =4 = + s0 ln ǫ d 4πǫ2
a b 2 s0 = χ[Σ] [Wabab Tr ˆk ] 180 − 240π ZΣ −
χ[Σ] is Euler number of Σ
Wabab is projection of Weyl tensor on subspace orthogonal to Σ, na, a = 1, 2 is a pair of normal vectors
a a 1 a ˆk = k γ νk , a = 1, 2 is trace-free ν ν − d 2 extrinsic curvature− of Σ
7 EE in d dimensions
In d dimensions compact entangling surface Σ is (d 2)-dimensional −
Logarithmic term s0 in entanglement entropy is given by integral over Σ of a polynomial in- variant constructed from Weyl tensor W ανβ, even number of covariant derivatives of Weyl ˆa tensor, extrinsic curvature k ν and projections a on normal vectors n .
If d is odd no such invariant exists so that
s0 = 0 if d = 2n + 1
8 In this talk:
What changes if manifold has boundaries?
9 Conformal boundary conditions
General (mixed) boundary condition is a com- bination of Robin and Dirichlet b.c.
( n+S)Π+ϕ ∂ = 0 , Π ϕ ∂ = 0 , Π++Π = 1 ∇ | M − | M −
10 Conformal scalar field in d di- mensions
Dirichlet b.c. (Π+ = 0)
φ ∂ = 0 | M
Conformal Robin b.c. (Π = 0) − (d 2) ( n + − K)φ ∂ = 0 ∇ 2(d 1) | M −
Remark: in d = 4 exists one more (complex) Robin b.c.
1 i 2 1 S = K 10Tr Kˆ , Kˆ ν = K ν γ νK 3 ± 10q − 3 for which (classical and quantum) theory is conformal
11 Dirac field in d = 4 dimensions
Π ψ ∂ = 0 , ( n + K/2)Π+ψ ∂ = 0 − | M ∇ | M
1 Π = (1 iγ n γ ), γ is chirality gamma ± 2 ± ∗ ∗ function
12 Integral Weyl anomaly
Variation of effective action under constant rescaling of metric
2σ ∂σW [e g ν]= T A≡ ZMd D E
For free fields integral Weyl anomaly reduces to computation of heat kernel coefficient Ad.
13 General structure
ν √g T ν g = a χ( )+ b √γI (W ) Md k k ZMd ZMd ˆ ˆ +bk′ √γJk(W, K)+ cn √γ n(K) , ∂ ∂ K Z Md Z Md
χ[ ] is Euler number of manifold , I (W ) Md Md k are conformal invariants constructed from the Weyl tensor, n(Kˆ) are polynomial of degree K (d 1) of the trace-free extrinsic curvature, − 1 K ν = K ν d 2γK is trace free extrinsic cur- − − σ vature of boundary; Kˆ ν e Kˆ ν if g ν σ → → e g ν.
14 Q: Does it mean that there are new conformal charges bn′ , cn?
A: we suggest that in appropriate normaliza- tion bn′ = bn and the corresponding boundary term Jk(W, Kˆ) is in fact the Hawking-Gibbons type term for the bulk action Ik(W ) cn are indeed new boundary conformal charges
15 Gibbons-Hawking type terms
Re-writing functional of curvature in a form linear in Riemann tensor
I = Uαβ νR Uαβ νV + F (V ) bulk αβ ν − αβ ν ZMd
In order to cancel normal derivatives of the metric variation on the boundary one should add a boundary term, αβ ν (0) Iboundary = U P − ∂ αβ ν Z Md (0) P = nαnνK n nνKα nαn K +n n Kαν αβ ν β − β − βν β
n is normal vector and K ν is extrinsic curva- ture of ∂ Md
Barvinsky-SS (95)
16 For a bulk invariant expressed in terms of Weyl tensor only,
I[W ]= Uαβ νW Uαβ νV + F (V ) αβ ν − αβ ν ZMd αβ ν U Pαβ ν − ∂ Z Md (0) 1 (0) (0) (0) P = P (gα P gανP g Pαν αβ ν αβ ν−d 2 βν − β − β − (0) (0) P +g Pα )+ (gα g gανg ) βν (d 1)(d 2) βν − βν − − (0) α α P ν = n n K + n n Kαν K ν n nνK αβ − − P (0) = 2K −
Pαβ ν has same symmetries as the Weyl tensor. α In particular, P αν = 0.
Pαβ ν can be expressed in terms of Kˆ ν
17 Examples
n n 1 1. Tr (W ) nTr (PW − ) − ∂ ZMd Z Md
2. Tr (W 2W ) 2 Tr (P 2W ) ∇ − ∂ ∇ ZMd Z Md
18 Integral Weyl anomaly in d = 4: anomaly of type A
First of all, bulk integral of E4 is supplemented by some boundary terms to form a topological invariant, the Euler number, 1 χ[ 4]= E4 M 32π2 ZM4 1 ν ν 1 (K Rn nν K R ν KRnn + KR −4π2 ∂ − − 2 Z M4 1 2 K3 + KTr K2 + Tr K3) −3 3
α β ν R nνn = R ανβn n and Rnn = R νn n
Dowker-Schofield (90)
Herzog-Huang-Jensen (2015)
19 Integral Weyl anomaly in d = 4: anomaly of type B
Gibbons-Hawking type boundary term:
Tr W 2 2 Tr (WP ) − ∂ ZM4 Z M4
Due to properties of Weyl tensor:
(0) ναβ Tr (WP ) = Tr (WP ) = 4W n nβKˆνα
20 Integral Weyl anomaly in d =4
a T = χ[ ] −180 M4 ZM4
b 2 ναβ + Tr W 8 W n nβKˆνα 1920π2 − ∂ ! ZM4 Z M4 c + Tr Kˆ 3 280π2 ∂ Z M4
For B-anomaly balance between bulk and bound- ary terms agrees with calculation for free fields of spin s=0,1/2, 1
Fursaev (2015) also Herzog-Huang-Jensen (2015)
21 Values of boundary charge c:
(Malmed (88), Dowker-Schofield (95), Fursaev (2015)) c = 1 for s = 0 (Dirichlet b.c.) c = 7/9 for s = 0 (Robin b.c.) c = 5 for s = 1/2 (mixed b.c.) c = 8 for s = 1 (absolute or relative b.c)
22 Local Weyl anomaly in d =6
T = = aE + b I + b I + b I + TD A 6 1 1 2 2 3 3 where E6 is the Euler density in d =6 and we defined
3 σρν αβ I1 = Tr 1(W )= Wα νβW Wσ ρ
3 ν σρ αβ I2 = Tr 2(W )= Wαβ W ν Wσρ
I = Tr (W 2W )+ Tr (WXW ) 3 ∇ 2 [ ] 6 X ν = X δν , X = 4R Rδ αβ [α β] ν ν − 5 ν
23 Integral Weyl anomaly in d =6
T = a χ[ ] ′ M6 ZM6
3 2 +b1 Tr 1W 3 Tr 1(PW ) − ∂ ! ZM6 Z M6
3 2 +b2 Tr 2W 3 Tr 2(PW ) − ∂ ! ZM6 Z M6 2 2 +b3[ Tr (W W ) 2 Tr (P W ) ∇ − ∂ ∇ ZM6 Z M6
+ Tr 2(WXW ) Tr 2(WQW )] − ∂ ZM6 Z M6 2 3 5 + c1Tr Kˆ Tr Kˆ + c2Tr Kˆ ∂ Z M6 two new boundary charges c1 and c2 there may exist additional invariant with deriva- tives of extrinsic curvature 24 Integral Weyl anomaly in odd di- mensions
Euler number of vanishes if d is odd Md
Euler number of boundary ∂ d may appear in integral anomaly M d = 3 :
c1 c2 2 T = χ[∂ 3]+ Tr Kˆ 96 M 256π ∂ ZM3 Z M3
(c1,c2):
( 1, 1) for scalar filed (Dirichlet b.c.) − (1, 1) for scalar field (conformal Robin b.c)
(0, 2) for Dirac field (mixed b.c.)
Remark: similar anomaly for defects Jensen- O’Bannon (2015)
25 Integral Weyl anomaly in d = 5
T = c χ[∂ ] 1 M5 ZM5 2 α β αβ + [c2Tr W +c3WαnβnW n n+c4Wnαβ Wn ∂ Z M5 α βν ˆ ˆ α β ˆ ˆ σ +c5W KαβK ν + c6W n nKασK β
+c (Tr Kˆ 2)2 + c Tr Kˆ 4 + c Tr (Kˆ Kˆ)] 7 8 9 D
is conformal operator acting on trace free D symmetric tensor in 4 dimensions values of ck for conformal scalar field: work in progress with Clement Berthiere
26 Entanglement entropy: d =3 (recent work with Fursaev)
Renyi entropy
S(n) c(n)L/ǫ ln(ǫ)s(n) ≃ − nA (1) A (n) s(n) = η 3 − 3 n 1 −
A3(n) is heat kernel coefficient on replica man- ifold n M
27 Consider = R2 L, L is an interval with 2 M × end points P1 and P2
Entangling surface Σ = L, replica space n = M n L C ×
A (n)= A ( n) A (L) , 3 2 C × 1
1 2 A2( n)= (1 n ) C 12n − is the heat kernel coefficient on two-dimensional cone, and
1 A1(L)= tr χ, χ =Π+ Π 4 − − XPk
28 for scalar field
c n + 1 c s(n) = 1 , s(n=1) = 1 48 n 24 XP XP for Dirac field
s(n) = 0 , s(n=1) = 0 INTERESTING PREDICTION: dependence on angle between entangling sur- face Σ and boundary ∂M cos α = (n,t), n normal vector to ∂ , M3 t tangent vector to Σ.
Assume that the bulk n contains a conical M singularity then: scalar curvature of the boundary
Rˆ 4π cos α (1 n) , n 1 ∂ n ≃ − → Z M and extrinsic curvature of the boundary
K2 Tr K2 8π(1 n)f(α) , ∂ n ≃ ∂ n ≃ − Z M Z M 1 sin2 α f(α)= (1 + 2 cos2 α + 5cos4 α) −32 cos α
29 OTHER DIMENSIONS
P =Σ ∂ dim(P )= d 3 ∩ Md − pa ,a = 1, 2 normal vectors to P in ∂ d ˆa M k ν is respective extrinsic curvature of P
ˆ α β ˆ Kab = pa pb Kαβ d = 3 : dim(P ) = 0 s (P ) 0 ∼ P P d = 4 : dim(P ) = 1 s (P ) Kˆaa 0 ∼ P R d = 5 : dim(P ) = 2 possible terms in s0(P ): χ(P ) , Wnana , Wabab , 2 2 (Kˆaa) , KˆabKˆab , tr ˆk and terms with two derivatives of extrinsic curvature
30 RESUME in presence of boundaries integral Weyl anomaly is modified by boundary terms boundary terms for B-anomaly are of Gibbons- Hawking type additional new boundary charges in odd dimensions integral Weyl anomaly is non-vanishing (!) and is entirely due to bound- ary terms if intersection of entangling surface and bound- ary is P then there appear new contributions to EE (and RE) due to P in odd dimensions log term in EE (and RE) is non-vanishing (!) and is entirely due to P
31 SOME OPEN QUESTIONS
1. how derive boundary charges from n-point correlation functions in CFT?
2. what is holographic description of boundary terms in anomaly and in EE? (work in progress with Amin Astaneh)
32 THANK YOU!
33